Do domestic partners have the same rights as spouses?

Absolutely not.  A domestic partnership is almost always going to be something less than a marriage;  it is why the same-sex-marriage issue is as important as it is.  In Florida, the law itself does not give any rights to you by being in a domestic partnership.

However, if you and your partner have a properly drawn domestic partnership agreement, the law will usually enforce that;  this can cover issues like support and property division in the event of a break-up.  But the existence of the agreement, itself, will not give you the right to inherit, and you may well need to do something separate, like a living will, to allow your partner to have access to your health care information and to decide for you if you are unable.